If you've been doing it for a month and have noticed that you're getting slimmer then you're obviously doing something right. Getting in shape is never a quick fix, more like a permanent change of lifestyle. And yes muscle weigh a bit more than fat, so you might initially get slimmer without getting lighter.

Get yourself a measuring tape and start measuring your waist and chest circumferences, that's usually more reliable than checking your weight (although you want to continue doing that too). If your waist is getting smaller, then you're losing fat, and if your chest is getting bigger (without your waist doing the same thing) then you're gaining muscles.

you have to make your workout a priority, I have a crazy schedule and still manage to train 2-3 times a day, 24 hours in a day. You have mention that you are trying to cut fat and carbs, be careful, eat good fat and good carbs, you do need carbs when trainning mma. you eat 5 times a day, I suggest you eat 6-8 meals a day. breakfest is very important especially if you want to cut weight, as soon as you wake up drink water and eat, it will kickstart your metabolism. Drink alot of water, helped me alot.